Vapor pressure is property of liquid and partial pressure is property of a gas in a mixture.
The normal boiling point for the pure component in liquid phase is the temperature at which vapor pressure exerted by the liquid becomes equal to 1 atm.
In a gas mixture, as long as partial pressure of a component remains less than its liquid phase vapor pressure at the temperature of the mixture, the component will not condense. Condensation starts only from the point when $PP=VP$ and starting from the case where $PP>VP$, condensation continues only till $PP=VP$. When $PP<VP$, condensation stops.
Air at 1 bar total pressure will have 0.21 bar partial pressure of $\ce{O2}$ and say it is at -183C (the normal boiling point of $\ce{O2}$). Vapor pressure of liquid $\ce{O2}$ at this temperature is 1.01 bar, but partial pressure is 0.21 bar which is less than 1.01 bar so $\ce{O2}$ should not condense. But if Air is at 5 bar total pressure, this will have $0.21\times 5=1.05$ bar partial pressure of $\ce{O2}$ and this air is at -183C then $\ce{O2}$ will condense till its partial pressure has gone below 1.01 bar.
Vapor pressure is independent of composition in liquid phase but dependent on temperature. As temperature increases, the vapor pressure of a liquid component increases. Vapor pressure does not depend on gas phase mixture properties.
At a given temperature, the component having the higher vapour pressure in liquid phase is the more volatile components and will have lower boiling point.